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Read the German Wikipedia article on the subject which shows German signs in short tons (non-metric unit): https://de.wikipedia.org/wiki/Militärische_Lastenklasse

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The class numbers only roughly correspond to weight, so you could just as well consider them metric tons (1000 kg).

You definitely can't consider them metric tons. If the bridge you're about to cross has a maximum advertised load of 120 sh tn and you consider it safe to drive your 120 m tn vehicle over the bridge on the basis of that you're going to be putting a 132.3 m tn load on the bridge, or roughly 10% more than it's rated for.

As I said above, these signs do not show a literal maximum weight. They show a load class.

Right, I was overly simplifying. I meant to say a 120 sh ton MLC-calculated weight, not a vehicle that's exactly 120 tons since as the article goes into the class needs to take into account axle count etc.

What I don't understand is your claim that "you could just as well consider them metric tons". These are short tons with caveats, i.e. the eventual number depends on more than just the raw weight, but the raw weight is one aspect of the calculation.

So if you were to make that calculation on the basis of metric instead of lbs how aren't you going to introduce something like a 10% error in the MLC you come up with?

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